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to the psychologist is not the theorem but the circumstances/
For fifteen days I struggled to prove that no functions
analogous to those I have since called Fuchsian functions
could exist; I was then very ignorant. Every day I sat
down at my work table where I spent an hour or two; I
tried a great number of combinations and arrived at no
result. One evening, contrary to my custom, I took black
coffee; I could not go to sleep; ideas swarmed up in clouds;
I sensed them clashing until, to put it so, a pair would hook
together to form a stable combination. By morning I had
established the existence of a class of Fuchsian functions,
those derived from the hypergeometric series. I had only to
write up the results, which took me a few hours.
Nest I wished to represent these functions by the quo-
tient of two series; this idea was perfectly conscious and
thought out; analogy with elliptic functions guided me. I
asked myself what must be the properties of these series if
they existed, and without diiTiculty I constructed the series
which I called thetafuchsian.
I then left Caen, where I was living at the time, to parti-
cipate in a geological trip sponsored by the School of
Mines. The exigencies of travel made me forget my mathe-
matical labours; reaching Coutances we took a bus for
some excursion or another. The instant I put ray foot on
the step the idea came to me, apparently with nothing
whatever in my previous thoughts having prepared me for
it, that the transformations which I had used to define
Fuchsian functions were identical with those of non-
Euclidean geometry* I did not make the verification: I
should not have had the time, because once in the bus I
resumed an interrupted conversation; but I felt an instant
and complete certainty. On returning to Caen, I verified
the result at my leisure to satisfy my conscience.
I then undertook the study of certain arithmetical ques-
tions without much apparent success and without suspect-
ing that such matters could have the slightest connexion
with my previous studies. Disgusted at my lack of success,
I went to spend a few days at the seaside and thought of
something else. One day, while walking along the cliffs,
the idea came to me, again with the same characteristics
of brevity, suddenness, and immediate certainty, that the